Complex physical processes are often more easily understood by developing a simplified model to describe the relationship between the variables defining the process. The diffusion of solute into a solvent is such a process. Use of the modeling approach is therefore particularly applicable to a surface acoustic wave (SAW) device in which a chemical vapor is absorbed by a polymeric coating. The polymeric coating applied to a surface of the SAW device is selected to absorb a desired range of chemical vapors. As a chemical vapor diffuses into the polymeric coating, it changes the mass of the coating, and thus correspondingly changes the phase and delay of a Rayleigh surface wave propagating between electrodes disposed at each end of the polymer coated surface. By monitoring the change in resonant frequency of an amplifier connected with the SAW device in its feedback loop, the diffusion of the chemical vapor into the polymer coating and its steady-state level within the coating can be determined. Chemical detectors have been built that comprise an array of SAW devices, each device having a different polymer coating exhibiting a unique characteristic solubility for a specific chemical vapor or class of chemical vapors to which the chemical detector might be exposed. The chemical detector is thus usable to identify a specific chemical substance, based on the relative steady-state levels of the chemical substance in the different polymer coatings of each SAW device in the array.
A simple model for the diffusion of a chemical vapor into the polymer coating of a SAW device has been proposed, corresponding to an ideal one-dimensional diffusion. In the general case, a diffusion process is typically described by the equation: ##EQU1## where f(t) represents a signal that varies as diffusion proceeds;
C.sub.o is the steady-state level of the signal f(t); PA1 C.sub.n are the C.sub.1 . . . C.sub..infin. components of the diffusion process; PA1 .tau..sub.n are the reciprocals of the decay times for each of the n components, ordered by magnitude, with .tau..sub.1 being the largest; and PA1 t is the elapsed time for the diffusion.
Applying Equation 1 to a SAW device used as a chemical detector, the value of f(t) defines the resonant frequency of the SAW device at time t. For a given concentration of a chemical substance in the SAW sensor coating, C.sub.o represents the resonant frequency for the steady-state level of that substance within the coating. The summation of the components on the right of Equation 1 defines the transition from an earlier steady-state level to C.sub.o, following a change in the ambient concentration of the chemical substance.
Based on the simplified model described by Equation 1, the resonant frequency associated with the SAW device should change exponentially, starting at an initial time, t.sub.0, when it is first exposed to a chemical vapor, until at a time, t.sub.0 +3.tau..sub.1, when the frequency is within a few percent of its new steady-state level, C.sub.o. Unfortunately, although the experimental data for a diffusion process may visually appear to fit this simple exponential form when graphically illustrated, such data in fact deviate sufficiently from the model and the equation to create problems in predicting the time constant for diffusion, .tau., and the steady-state level, C, of the chemical substance in the coating of a SAW device.
The ability to predict the time for diffusion and the steady-state level of a chemical vapor into a polymer coating of a SAW device (represented by its resonant frequency) is particularly important in applications where the SAW device may be used to detect very dilute concentrations of potentially harmful chemical vapors, or alternatively, where the concentration of chemical vapor is so great that it would quickly saturate the SAW device, rendering it unusable for detecting other chemical vapors for an extended period of time. Clearly, the detection and identification of harmful substances must be completed as rapidly as possible, before personnel exposed to the chemical substance are critically affected. Since, for a very dilute ambient concentration of a chemical vapor, it may require up to 15 minutes for the diffusion of the chemical vapor into the polymer coating of a SAW device to asymptotically approach a steady-state level, early prediction of that parameter, e.g., within 3-4 minutes, may correspondingly greatly reduce the time required for identification of a potentially harmful substance. In addition, since various chemical substances are more effectively characterized by both their steady-state level and the time constant for diffusion of the substances into a specific polymer coating, early prediction of the time constant for diffusion is equally important. Use of these two predicted values to identify a chemical substance and to determine its ambient concentration is the subject of a commonly assigned U.S. Pat. No. 4,895,017, entitled, "Apparatus and Method For Early Detection and Identification of Dilute Chemical Vapors," filed Jan. 23, 1989.
The simple exponential model discussed above fails to deal with an initial negative rate of diffusion that is reflected in the resonant frequency data produced by a SAW device exposed to a chemical vapor. In addition, the above model is based on the assumption that the exponential function does not change in time. The actual experimental data developed from SAW devices show that the exponential function varies in time and is not "well behaved" in its initial change when first exposed to a chemical vapor. It is possible to predict the time for diffusion and the asymptote of the diffusion process using a Kalman filter technique, but this approach may not provide optimum speed and accuracy.
Although specifically developed for use with the SAW device, a method for determining a steady-state level and a time to achieve the steady-state level in accordance with the present invention is generally applicable for use with any time-varying exponential process, e.g., to predict the time and final charge on a capacitor as current flows into it through a solid-state switch, or to predict parameters relating to the flow of heat into or out of an object. Complicating any of these time varying exponential processes are the random variations in experimental data from which parameters characterizing the processes must be developed. The problem arises because datum developed experimentally may deviate substantially from the nominal best fit time varying exponential curve. It is therefore relatively difficult to use traditional techniques to fit a time varying exponential curve to such data.
In consideration of the above problems, it is an object of the present invention to provide a method for predicting an asymptote of a signal that changes according to a time varying exponential curve. It is a further object of this invention to predict the characteristic time required for the signal to substantially approach an asymptotic value. Yet a further object is to map preliminary diffusion data into an asymptote/time domain, permitting the data to be fit to a time varying exponential curve, from which the asymptote and time for diffusion may be predicted. These and other objects and advantages of the present invention should be apparent from the attached drawings and the Disclosure of the Preferred Embodiment that follows.